DEFINITION of the DERIVATIVE
We define the derivative as follows:
The fraction:
is called the "difference quotient" and really represents the slope of a line segment drawn between the point (x,y) and the next point produced with x+h .
The limit as "h approaches zero" is the way that we make the distance between the two points smaller and smaller and smaller. Eventually, the distance between the points is indistinguishable.
In this way, smaller and smaller line segments are produced, literally sliding down to a single point on the curve. When a line touches a curve at a single point, we call the line a tangent.
For this reason, we say that the derivative produces the slope of a line tangent to a curve at x.
EXAMPLE using conjugates
Use the definition of the derivative to find the slope of a line tangent to the following curve at x = 2
First use the definition of the derivative.
Notice the two fractions in the numerator.
Begin by factoring 2 and then writing the two separate fractions as one fraction with a common denominator.
We will now get rid of the radicals in the numerator and cancel a common factor. As you will see, the common factor will turn out to be "h".
We do this by multiplying the top and bottom of the fraction by the conjugate of the numerator.
If we have two terms a + b, the conjugate is a - b.
The conjugate of the numerator in our problem is:
We multiply the top and the bottom of the difference quotient by this conjugate.
This will look pretty scary, but stay calm and trust your algebra.
Remember, our original fraction was being divided by "h":
We can now take the limit of this new fraction by using all of the limit laws.
Remember: the limit of a quotient is the quotient of the limits; the limit of a constant is the constant; the limit of a product is the product of the limits; and the limit of a sum is the sum of the limits. Whew!
In order to find the slope of a line tangent to the curve at x = 2, replace the x by "2" and do the arithmetic.
The slope of a line tangent to
at x = 2 is